论文标题
粒子方法和基于量化的方案，用于模拟McKean-Vlasov方程
Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation
论文作者
论文摘要
在本文中，我们研究了McKean-Vlasov方程的三个数值方案\ [\ begin {cases} \; dx_t = b（t，x_t，μ_t，μ_t）\，dt+σ（t，x_t，x_t，μ__t，μ__t，μ_t，μ_t，μ_t） \ text {是} x_t，\ end {case} \]的概率分布，其中$ x_0 $是已知的随机变量。在lipschitz连续性的假设下，系数$ b $和$σ$，我们的第一个结果证明了粒子方法相对于Wasserstein距离的收敛速率，该方法扩展了以前在一维环境中建立的工作[BT97]。在第二部分中，我们介绍和分析了两个基于量化的方案，包括Vlasov设置中的递归量化方案（确定性方案）和混合粒子定量方案（随机方案，灵感来自$ K $ -MEANS群集）。在本文的末尾进行了两个示例：汉堡方程和菲茨胡格 - 纳古莫神经元的网络在维度3中。
In this paper, we study three numerical schemes for the McKean-Vlasov equation \[\begin{cases} \;dX_t=b(t, X_t, μ_t) \, dt+σ(t, X_t, μ_t) \, dB_t,\: \\ \;\forall\, t\in[0,T],\;μ_t \text{ is the probability distribution of }X_t, \end{cases}\] where $X_0$ is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients $b$ and $σ$, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends a previous work [BT97] established in one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme, inspired by the $K$-means clustering). Two examples are simulated at the end of this paper: Burger's equation and the network of FitzHugh-Nagumo neurons in dimension 3.
